Why can't the discount rate be lower than the growth rate in terminal value?

What is the theoretical reason for it.

Thanks.

### Ways to Calculate Terminal Value

Terminal value is an important part in determining company valuation. Before digging in to the theoretical explanation to the above question, here's a quick review of the calculation. Depending on various factors, you may want to use an exit multiple or perpetual growth method, such as the Gordon Growth Model for determining terminal value in a DCF model.

• Perpetual Growth: Use when company is in its long-term, mature growth phase
• Terminal Value = Last Year Free Cash Flow x ((1 + Terminal Growth Rate) / (WACC - Terminal Growth Rate))
• Exit Multiple: Use when company is not yet in steady growth phase or when market has a good idea of acquisition value (ex: LBO)

For more information on how to find your growth rate and discount rate, check out these posts:

Remember, no matter what formula and inputs you use, it is just an approximation or attempt to model a complex real world process.

### How Growth Rate and Discount Rate Impact Terminal Value Formula

From a simple mathematical perspective, the growth rate can't be higher than the discount rate because it would give you a negative terminal value. From a theoretical perspective, Certified Investment Banking Professional - 1st Year Associate @jhoratio explains:

Growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it.

Um, the discount rate is higher than the growth rate buddy, not the other way around.

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The model does not work because it would give you a negative number (impossible). You would need to use EBITDA and an exit multiple to find terminal value.

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Uh, it has to be, otherwise you get negative terminal value.

n - terminal year, r - discount rate, g - growth rate

Perpetuity PV = (FCF @ n+1) / (r - g)

If r <= g, PV <= 0 (or really it's probably infinite, but the formula gives you a negative value).

Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

nick_123:

Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

• 1 BobbyLight:
nick_123:

Sorry, I meant the other way around (growth rate cannot exceed the discount rate).

I understand that it gives you a negative number in the formula, but what is the theoretical reason for this?

Why can't we expect a company to grow faster than its discount rate in the future?

A company's long-term growth rate isn't going to surpass the amount required for investors of all securities to take on the risk. Just think about it, first off it's a projection based on assumptions, so it's not a reflection of actual returns or growth.

Hmmm, I'm still not seeing it. Why wouldn't a company's growth rate be able to surpass the required return of investors?

After all, growth and risk are driven by different variables.

nick and Bobby, you were saying the same thing.

giants92:

nick and Bobby, you were saying the same thing.

No, the original post has been corrected.

If a company were to grow faster than the expected rate of return in perpetuity, in effect growing faster than the market itself, then the company would be on pace to eventually become larger than the entire market. Impossible.

Best Response

Nick, growth rates can exceed the cost of capital for very short periods of time, but we're talking about a growth rate IN PERPETUITY here. This is kind of like asking, "why don't trees just keep growing past the clouds?" or "why can't a stock be worth less than zero?" You're just playing with numbers here and have forgotten the underlying reality. Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! Seriously, unless you think this is a likely scenario, it just simply cannot be that the growth rate exceeds the risk. Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

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With an answer like that, you should be working at a hedge fund.

Absolute truths don't exist... celebrated opinions do.

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jhoratio:

Any company whose growth rate exceeds the required rate of return would a) be a riskless arbitrage and b) attract all the money in the world to invest in it. The company would eventually become the entire economy with every human being on earth working for it. Wow! Talk about a conglomerate! [...] Remember all these formulas are just mathematical APPROXIMATIONS of incredibly complex real world processes. Don't let the tail wag the dog.

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It's a convergent geometric series (infinite series that converges to a finite sum) as long as growth < r. However, it becomes a divergent geometric series (diverges to infinity) when g > r. Hence, the model blows up when g > r.

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Perpetuity Growth rate higher than required rate of return = a bubble that will never burst.. like puff in the microwave...

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Banker: "Where's me Bonus?"
Yuppie: "Whadya mean I have to actually work?"
Fox Rock Mum: "Lidl's the place to be seen in now!"
Cowen: "It's not my fault that me and my party are complete f**k-ups - it's the recession silly!"

If you have any perpetual yearly cash flow that grows at a rate greater than the discount rate, your NPV will be infinite. Think about it this way - every future year's present value will be greater than the previous - because your cash flow is growing faster than you can discount it - and thus you'll will not obtain a finite net present value.

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Question:

i have a project planned to start on 1 jan 2010, is able to produce revenue cash flow starting from 1million on 1 jan 2011 and this is expected to grow at 13% per year until 1 jan 2020.

The initial cost of the project is 20million. Beta: 1.1, risk free rate: 1.7%, risk premium is 9.5%.

So the cost of capital should be 0.017 + 1.1(0.095) = 0.1215

but it is lower than the growth rate of 13%.

is my calculation wrong?

No since your growth rate is not perpetual.

So basically the terminal value can't be negative ??

This is a purely mathematical question, it can be answered with zero economic reasoning (although you can go further with it to give the economic interpretation like the guys went here). The sum of perpetual "cash flows" (or whatever) growing at a constant rate and being discounted through time only converges to a value (the "V = F/(k-g)" formula) if the growth rate is lower than the discount rate. If it is higher, this value (the sum of the cash flows, not the formula, which is conditional to this specific case: k > g) is infinite (as all the terms of the sum are higher than one, and the sum is infinite). It can be negative if the numerator is negative (the "cash flows"), and if the growth rate is higher than the discount rate, then it is infinitely negative.

"Never believe in anything until it has been officially denied"

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So in a 2 stage model, can the growth rate in the initial short-term high growth period exceed the firm's WACC, as long as the perpetual growth rate is < WACC?

absolutely, imo you can model any scenario you like and can defend. however these assumptions (as always) need to be reasonable, meaning that the stages should not have abrupt changes in the pace of growth. A way to prevent this is adding a mechanical appendix to the initial forecast (say 6 years initial forecast + 4 years appendix), where growth rates are slowly converging to the perpetuity growth rate assumed.

i believe this method often increases the validity of a model

This response is from a purely conceptual economic point of view (ignoring all of the other valid practical arguments like trees don't grow to the sky and mathematical arguments around divergent series).

If you think about a discount rate as a required rate of return, this becomes an easier question to understand.

Roughly speaking, a security's return / discount rate =
1. yield plus
2. operations growth (EBITDA, FCF, whatever) plus
3. changes in multiple.

Re #3, in perpetuity, you don't get any of the effects of changes in multiple. So we just have yield + operations growth.

Re #1, if we're talking about a strictly positive FCF business, the yield has to be positive.

Therefore, your growth rate must be less than your overall rate of return since your rate of return should take into account both growth and positive yield.  